Finite Volume Element Methods for Two-Dimensional Three-Temperature Radiation Diffusion Equations

Abstract

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multi-material system and explain the exchange of energy among electrons, ions and photons. Their highly nonlinear, strong discontinuous and tightly coupled phenomena always make the numerical solution of such equations extremely challenging. In this paper, we construct two finite volume element schemes both satisfying the discrete conservation property. One of them can well preserve the positivity of analytical solutions, while the other one does not satisfy this property. To fix this defect, two as repair techniques are designed. In addition, as the numerical simulation of 2-D 3-T equations is very time consuming, we also devise a mesh adaptation algorithm to reduce the cost. Numerical results show that these new methods are practical and efficient in solving this kind of problems.

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

“…Since the parameters ρ, c vα (α = e, i, r), ω eα (α = i, r) are possibly discontinuous across the interfaces of different material areas, the approximate values of these parameters inside the dual cells need to be considered specially. Here we suggest the strategy in [7] to deal with these parameters. Let η be a possibly discontinuous parameter, then the restriction of η on the dual cell K * ν is given by…”

“…The first scheme suggested in [51] (resp. [7]) for twodimensional two-temperature (resp. two-dimensional three-temperature) radiation diffusion problems is denoted as FVEM-2T (resp.…”

“…In [47], two substructuring nonoverlapping domain decomposition preconditioners were employed to solve the SFVE discretization of 2-D 3-T radiation diffusion equations with strongly discontinuous coefficients. In [7], two finite volume element schemes were constructed on triangular meshes. The authors in [31] adopted the freezing coefficient method to linearize the nonlinear equations and then solved the resulting equations by Raviart-Thomas mixed finite element method.…”

“…However, numerical schemes mentioned in the previous paragraph do not possess the positivity-preserving property unconditionally. The first scheme in [7] can preserve the positivity of the solutions if and only if the maximal interior angle of the primary partition is not greater than π/2, while the second scheme adopted two post-processing techniques, i.e., global repair technique [22,41] and cutoff method [21], to keep the positivity-preserving property, which may possibly destroy the accuracy and the energy conservation. Hence, for 2-D 3-T equations on general polygonal meshes, it is interesting to design a strictly positivity-preserving finite volume scheme that can keep both the energy conservation and the second-order accuracy.…”

“…The first and fifth features are the main differences from the ones in [9], while the rest ones make it most attractive than those in [7,31,51]. The rest of this paper is organized as follows.…”